I've figured out probably the best way to integrate OBS and OpenLP. It works whether OpenLP is running locally on the same computer, and also if OpenLP is running on a separate computer (because it's all based on browser sources).
There's two components:
Dock the OpenLP web controller into OBS as a Custom Browser Dock.Add OpenLP lyrics to OBS as a Browser Source, by using a simple custom Stage View. No greenscreening is needed.
It only takes 5 minutes to set up, and I've got a full video below showing the steps and some tips.
Finally, to send PowerPoint or video to OBS, you can use the standard Screen Capture source (if OpenLP is local) or NDI (if OpenLP is on a different computer).
Let me know what you think!